Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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A certain inverse limit

Let $p$ be an odd prime and $n$ a positive integer. Let $\zeta_{p^{n+1}}$ be a primitive $p^{n+1}$-th root of unity. It can be shown that $Gal(\mathbb Q(\zeta_{p^{n+1}})/\mathbb Q)\cong (\mathbb ...
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Computing the degree of a Galois extension.

Let $K = \Bbb{Q}(3^{1/5}, 3^{1/3})$. Compute $[K: \Bbb{Q}]$ (degree of the extention). Find $\alpha$ (not unique) such that $F=\Bbb{Q}(3^{1/5}, 3^{1/3} , \alpha)$ is the smallest Galois extension ...
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Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$

Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$. My justification for this question is as follows; Suppose $F(\alpha^2)\subsetneq F(\alpha)$, we have $F \subsetneq F(\alpha^2) ...
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Generalization of Kummer isomorphism?

Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
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Finding the Galois group of $x^6-x^5+x^4-x^3+x^2-x+1$ over $\Bbb{Q}$

I was trying to find the Galois group of $x^6-x^5+x^4-x^3+x^2-x+1$ over $\Bbb{Q}$, but I'm confused, because I want to find the splitting field for this polynomial... but I have to find the roots ...
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Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of ...
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Can one trisect the angle $\theta=\arccos(-12/17)$?

I think this is not possible, and below is my proof. Proof: Using the angle tripling formula: $$\cos(3\theta) = 4\cos^3 (\theta)−3\cos(\theta) \ \to \ \cos(\theta) = 4\cos^3 (\theta/3) ...
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Prove that the angle $\theta=\arccos(-12/17)$ is constructible using ruler and compass.

Could I just do this? Proof: if we want to show $\arccos\left(\frac{-12}{15}\right)$ is constructible, can't I just say, take $x_0=\cos(\theta)=-\frac{12}{17}$ implies $17x_0+12=0$ which says that ...
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Can one trisect $\arccos(6/7)$?

Is this proof correct? Proof: Here $\theta = \arccos(6/7)$. Now to show we can't trisect $\theta$, we show that $\theta/3$ is not constructible by finding the irreducible polynomial in $\mathbb ...
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Applying Extended Euclidean Algorithm for Galois Field to Find Multiplicative Inverse

I was trying to apply the Extended Euclidean Algorithm for Galois Field. Among the many resources available, I found the methodology outlined in this document easy to grasp. The above works fine when ...
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Splitting field of resolvent equals that of $f$

Lemma: Let $\Psi \in k[X_1,...,X_n]=:B$ be s.t. $stab_{S_n}(\Psi)=H \subset S_n$, $S_n/H=\{ \Psi, t_2 \Psi,..., t_e\Psi \}$, $\Delta_\Psi$ the discriminant of $L_\Psi:=\prod_{i=1}^e (X-t_i \Psi)$, and ...
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Splitting field of $x^m - 1$ over $\mathbb F_p$

I need to find the splitting field of a polynomial $x^m-1 \in\mathbb{F}_p[x]$. I know that if $(m,p)=1$ then the splitting field is $\mathbb{F}_p(z)$ where $z$ is primitive root of unity of order $m$. ...
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A morphism which fixes one root of an irreducible polynomial must also fix the others.

Let $E/K$ be a field extension, let $p(x)$ be an irreducible polynomial in $K[x]$ which splits in $E$ with roots $\alpha_1$, $\alpha_2$, etc., and let $\sigma$ be an automorphism of $E$ which fixes ...
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$p$-adic Ring extensions vs. “ordinary” Ring extensions

I read about inverse limits in this post, and found the example by Arturo Magidin quite interesting (his "approximate" solution of $x^2 = -1$ in $\mathbb Z$). By his construction we get a Ring which ...
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Looking at the intermediate fields of $\Bbb{Q}_7 = \Bbb{Q}(\omega)/\Bbb{Q}$ where $\omega = e^{i2\pi/7}$ .

From Basic Abstract Algebra (Robert Ash): The question that I'm concerned with is number 3, but I will write problems 1 and 2 as well, since they are all related... We now do a detailed analysis ...
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Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?

Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such ...
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source to learn Galois Theory

What kind of recommendations do you have for a very good source to learn Galois Theory? Is there any Atiyah-MacDonald-type book on Galois theory? What is your opinion on the chapters from Lang and ...
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Why do no prime ideals ramify in the extension $\mathbb{Q}(\sqrt{p }, \sqrt{q})/\mathbb{Q}(\sqrt{pq })$?

Let $p,q $ be odd integer primes, $p \equiv 1 \pmod 4$ and $q \equiv 3 \pmod 4$. $K = \mathbb{Q }[\sqrt{pq }]$, $L = \mathbb{Q}[\sqrt{p }, \sqrt{q} ]$. Why a prime ideal in $O_{K}$ doesn't ramify in ...
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Galois Properties of the Values of Modular Forms

Let $f \in S_0(N)$ be a normalized Hecke eigenform. It is well known that its coefficients are algebraic integers, and $f^\sigma$ lies in $S_0(N)$ for $\sigma \in G_{\mathbb{Q}}$. At CM points $z \in ...
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Galois Group of $x^{12}+x^{11}+\dots+x^2+x+1$

It seems I have forgot all the details about the group theory. Anyone knows what is the Galois Group of $x^{12}+x^{11}+\dots+x^2+x+1$ and is that solvable? Thanks.
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In this theorem from Lang's Algebra, what is the codomain of this map?

I need help understanding what the final sentence below means. Paraphrased from Lang's Algebra: Let $K/k$ be Galois and $F$ an arbitrary extension, $K,F$ subfields of some other field. Then $KF/F$ ...
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Infinite-dimensional extensions of $\mathbb Q$

I need help to solve the following exercise: Let $X$ be an indeterminate over $\mathbb Q$ (so a transcendental number) and consider the field extensions $\mathbb Q\subseteq \mathbb ...
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Trouble with a proof in Dummit and Foote

Let $K/F$ be a Galois field extension and let $\alpha \in K$. We would like to show that the roots of the minimal polynomial $m_\alpha(x)$ for $\alpha$ are the distinct Galois conjugates of $\alpha$ ...
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Compute $\mathrm{Aut}(S)$ of the ring $S=\mathbb{Q}[x]/(x^2)$

This problem once again is from a previous exam. The problem is to compute the group of automorphisms of the ring $S$ where $S=\mathbb{Q}[x]/(x^2)$ My thoughts: Well $x^2$ is reducible over ...
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Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
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Galois actions on extensions of algebraic function fields

Let $k$ be an algebraically closed field, $C/k$ and $C'/k$ be smooth projective curves, and $C'/k \rightarrow C/k$ be a $k$-morphism which is corresponding to the field extension $k(C) \hookrightarrow ...
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Describing a Galois group.

From Robert Ash (Basic Abstract Algebra) Suppose that $E=F(\alpha)$ is a finite Galois extension of $F$, where $\alpha$ is a root of the irreducible polynomial $f \in F[x]$. Assume that the roots ...
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Possibilities for $[KL:F]$ when $[K:F]=[L:F]$ is prime

Suppose $K/F$ and $L/F$ are extensions of $F$ (contained in some common field) of degree $p$, where $p$ is prime. Standard arguments show that $[KL:F]$ must be in $\{p,2p,\ldots,p^2\}$. But are all ...
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About the number of $K$-homomorphisms

Let $K\subseteq L\subseteq M$ be fields with $[L:K]=n$. I should prove that the number of $K$-homomorphisms from $L$ to $M$ (so field homomorphisms that fix pointwise $K$) is less or equal than $n$. ...
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Let E be the splitting field of $f(x)=x^4-10x^2+1$ over $\Bbb{Q}$. find $Gal(E/\Bbb{Q})$.

From Galois Theory (Rotman): Let E be the splitting field of $f(x)=x^4-10x^2+1$ over $\Bbb{Q}$. find $Gal(E/\Bbb{Q})$. The roots of $f(x)$ are $\sqrt{2}+\sqrt{3}$, $\sqrt{2}-\sqrt{3}$, ...
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Proof of Abel-Ruffini's theorem

From Galois Theory (Rotman): I wrote down the whole proof, but my question is only about the third paragraph. There exists a quantic polynomial $f(x) \in \Bbb{Q}[x]$ that is not solvable by ...
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What is the splitting field of $x^3 - \pi$?

What is the splitting field of $x^3 - \pi$? Is it $\mathbb R(\sqrt[3] \pi, \xi_3)$ or $\mathbb Q(\sqrt[3] \pi, \xi_3)$? (where $\xi_3$ denotes the third root of unity) It is a polynomial over ...
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$K(u,v)$ is a simple extension of fields if $u$ is separable

I have problems to prove the following statement. Let $K$ be a field and let $K(u,v)$ be an algebraic extension of $K$. If $u$ is separable over $K$ then $K(u,v)$ is a simple extension. (My ...
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presentation of the inertia group of $p$-adic fields

It is known (see here) that the absolute Galois group of a $p$-adic number field $K$ (ie a finite extension of $\mathbb Q_p$) is topologically finitely presented for any odd prime $p$. Is it also ...
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Splitting field of polynomial over $\mathbb{F}_3$

Let $K$ be the splitting field of the polynomial $(x^3 + x - 1)(x^4 + x - 1)$ over $\mathbb{F}_3$. How many elements does $K$ contain? What I've already done is factoring $(x^3 + x - 1)(x^4 + x - 1)$ ...
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Vector Space isomorphisms of $\mathbb{Q}(z)$ preserving the Galois group (where $z$ is a primitive third root of unity)

Take the field extension $\mathbb{Q}(z)$ where $z$ is a primitive third root of unity and consider the set $A$ of vector-space automorphisms of $\mathbb{Q}(z)$ so that for $T \in A$ the map $\phi ...
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Patterns in $GF(2)$ Polynomial division.

I am testing Prime polynomials in $GF(2)$ and have noticed a pattern that I hope will help. There's a calculator here if you want to familiarise yourself with polynomials over $GF(2)$. I am testing ...
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Degree of factor in a resolvent

Background and lemma first: Let $\Theta \in k[x_1,...,x_n]^H$ and $\theta$ be its evaluation to the roots of a fixed $n$:th degree polynomial in $k[x]$. Put $L(t) = \prod_{\sigma \in S_n//H} (t-\sigma ...
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Group action on an ultraproduct

Let $\mathcal U$ be an ultrafilter, say on $\mathbb N$, and suppose we have a set $X$ with a group action by a group $G$ (for instance, one can think of $X$ as an algebraically closed field and $G$ as ...
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Normal extension

Can anybody help me with this problem ? Let $P\in {\mathbb Q}[X]$ be an irreducible polynomial of degree $p$ prime and let $z_1, z_2, \ldots ,z_p\in {\mathbb C}$ be the roots of $P$. Suppose that ...
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What could be said about a field extension, if it's Galois Group is fixpointfree?

Let $L/K$ be a galois extension, and let the Galoisgroup be fixpoint-free, what could be said about the field extension.
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Cyclotomic Polynomials over $\mathbb Q$ and reduction modulo $p$.

Let $p$ be prime, and let $\pi : \mathbb Z \to \mathbb Z / (p\mathbb Z)$ be the canonical projection $\pi(z) = z + p\mathbb Z$. Define its extension $\pi : \mathbb Z[x] \to \mathbb Z/(p\mathbb Z)[x]$ ...
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Question about the Galois extension of a given field extension

Let $K=\mathbb{Q}(\omega)$ be given, where $\omega^3=1$. I want to know: (1) Whether there is a Galois extension $L/\mathbb{Q}$ containing $K$ such that $\mathrm{Gal}(L/\mathbb{Q})\cong\mathbb{Z}_4$? ...
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What's wrong here when I compute $\operatorname{Gal} (x^4 -2 / \mathbb{Q})$

Maybe that's a stupid question and I'm missing something very trivial. Let $f(x) = x^4 - 2$ and $\alpha_1 = \sqrt[4]2, \alpha_2 = -\sqrt[4]2, \alpha_3 = i\sqrt[4]2, \alpha_4 = -i\sqrt[4]2$ be its ...
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Solvability by radicals of an equation of prime degree

For which prime $p$ the equation $x^p-p^px+p=0$ is solvable by radicals? I don't know how to solve this for primes $p\neq 2$, so any help is welcome.
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Where does the minimal polynomial lie?

Let $\theta = a + b \sqrt{D_1} + c \sqrt{D_2} + d \sqrt{D_1 D_2}$, where $a,b,c,d,D_1,D_2$ are integers. Is there any reason to believe the minimal polynomial for $\theta$ over $\mathbb{Q}$ should be ...
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Proof of Galois' theorem that there exists a field of $p^n$ elements.

From Galois Theory (Rotman): For every prime p and every positive integer n, there exists a field having exactly $p^n$ elements. Proof. If there were a field K with $|K| = p^n = q$, then ...
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Lower Bound on Number of Wildly Ramified Extensions of $\mathbb{Q}_p$

Suppose $p$ is prime and $p$ divides $e$ divides $n$. Typically up to isomorphism there are a lot of wildly ramified extensions of $\mathbb{Q}_p$ which have degree $n$ and ramification index $e$. ...
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When a subfield of a splitting field is a splitting field

Look at the following proposition: Let $K\subset L\subset M$ be three fields. If $M$ is a splitting field over $K$ of a polinomial in $K[X]$ and moreover if for every $\sigma\in G=Gal(M/K)$ we ...
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Galois extension corresponding to $S_n$

Given a positive integer $n$. How to find a Galois extension $K/F$ such that $Gal(K/F)=S_n$? With the restriction $F=\mathbb{Q}$, this is the Inverse Galois Problem. But if we are allowed to choose ...